Talk:Largest known prime number

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2^43112609 − 1 was nominated for deletion. The discussion was closed on 04 December 2011 with a consensus to merge. Its contents were merged into Largest known prime number. The original page is now a redirect to this page. For the contribution history and old versions of the redirected article, please see its history; for its talk page, see here.

M₁₃ a.k.a, 8191

In the history section, the first number of the list, under Discoverer lists ˝Anonymous˝ but fr:8 191 cites Abû Tahir Ismail al-Mardini as the discoverer. Should this change— Preceding unsigned comment added by Pqnlrn (talk • contribs) 15:46, 25 July 2019 (UTC)[reply]

6700417

Well, it seems Euler never actually said in print that it was prime; however, he had already done over half the work to prove it so, and apparently some experts believe he knew it anyway for the reason that the remaining work would not have taken him very long (it amounts to six trial divisions). (But then, why didn't he include it in the table of large primes he prepared in 1762?) So I have added a disclaimer about this. Double sharp (talk) 10:33, 18 April 2020 (UTC)[reply]

Moore's law

The chart showing increase in number of digits suggests that Moore's law is at work here. The connection is noted in Introduction to Cryptography]. SpinningSpark 17:19, 19 June 2020 (UTC)[reply]

Incorrect decimal expansion for 180×(M₁₂₇)²+1

I noticed that the decimal expansion entry for the prime number with the notation 180×(M₁₂₇)²+1 is currently:
5210644015679228794060694325390955853397149309953825381775591280356090833797121
From what I can tell, this number is neither prime nor what the notation should expand to.

The correct number should be:
5210644015679228794060694325390955853335898483908056458352183851018372555735221

Could someone double check this before editing the page? Aerdil (talk) 07:13, 27 June 2020 (UTC)[reply]

@Aerdil: Well spotted. I have fixed it.[1] Thanks. The wrong number was added by Putian Ye in [2]. It falsely computed M₁₂₇ as 2¹²⁷ instead of 2¹²⁷-1. I have checked all other numbers added in the edit and they are correct. PrimeHunter (talk) 08:02, 27 June 2020 (UTC)[reply]

I have spotted this non-prime at d:Q90570083 (then here). Fixed. Regards, —Mykhal (talk) 21:53, 31 December 2021 (UTC)[reply]

But Is There a Largest Prime Number in Theory?

I came to this article because I wanted to know the answer to the question: is there a proof that a largest prime number does or does not exist. I don't see that question addressed at all. I think it should be included in the article, is there a proof one way or another or is it an open question? --MadScientistX11 (talk) 04:30, 12 February 2021 (UTC)[reply]

I think the answer to my question is that Euclid proved there is no largest prime: Euclid prime proof. I think it would be a good idea to add that to the article. --MadScientistX11 (talk) 04:32, 12 February 2021 (UTC)[reply]

It's already in the second paragraph of the lead. PrimeHunter (talk) 22:05, 23 April 2021 (UTC)[reply]

Largest Probable Prime

The 5794777-repdigit 11111111...11111111 found Apr 20, 2021 by Ryan Propper & Sergey Batalov deserves to listed, but with a pointer to the Probable Prime page. EdPeggJr (talk) 04:30, 23 April 2021 (UTC)[reply]

Wikipedia:Articles for deletion/Titanic prime and the large prime number article proposal

D.Lazard has a proposal to bring together coverage of the search for large prime numbers under one article, large prime number, and has suggested that this article might provide the best starting point from which to build that article. I think it might be beneficial if (i) more editors weighed in at the AfD and (ii) we talked in more detail about how all of this material might best be organised. — Charles Stewart (talk) 08:41, 14 May 2021 (UTC)[reply]

I agree, Titanic primes should be de-listed since it overemphasizes the Lucas-Lehmer test. The new page should list largest primes proven by different primality tests. I recently added the largest probable prime R8177207 to this page with the distinction that it's not proven. 98.226.112.251 (talk) 14:57, 14 May 2021 (UTC)[reply]

This page semms like part of a large prime number, so it seems like we can already move this page to a large prime number. That may make the merge discussion easier.--SilverMatsu (talk) 05:28, 15 May 2021 (UTC)[reply]

Numbers of digits in this prime number

The power of 2 is 82589933 so By math,we can say n=8258993 The number of digit = 3n+1

                                   = 3×8258993 + 1
                                   = 24776979 + 1
                                   = 24776970 digits

So, 24776979 exactly digits in this prime number This is wrong that the numbers of digits in this prime number is 24862048.

If you want to any thing about this then reply. Thank you 2402:3A80:1F8C:9BC5:0:0:6C0:35A1 (talk) 18:40, 17 November 2021 (UTC)[reply]

You have several errors, starting with omitting an ending "3" in 82589933. The correct formula is ceil(n × log₁₀(2)) = ceil(82589933 × 0.3010299956...) = 24862048. The article is right. PrimeHunter (talk) 19:04, 17 November 2021 (UTC)[reply]

Definition of "known"

This article doesn't have any explanation of what it means for a prime number to be classified as "known". I think this should be fixed. Mexicochina (talk) 03:37, 23 December 2023 (UTC)[reply]

When the meaning of "known" primes is discussed at all in sources, it's about what constitutes a valid primality proof. Your edits show your concern is what it means for the actual number to be "known" when you have some formula or description of the number. That's a complete non-issue in sources so we shouldn't make it an issue. Quoting from my edit summaries: Reliable sources agree which primes are known without formally defining it. Wikipedia is based on sources and shouldn't invent a definition. Sources apparently haven't found a need to formalize it but everybody agrees that a decimal/binary expansion or simple formula like 2^p-1 is a known prime while a prime search algorithm expressed as a formula with a humongous number of steps is not a known prime. With currently known primality proving methods there is no gray zone between such clear cases so we have no good reason to make it sound problematic. Our main reliable source is titled Largest Known Primes. I have a website http://primerecords.dk and have personally found numerous prime records during many years and have participated in discussions with other record setters at many websites. When reliable sources agree what the largest known primes are, we can just list the primes with sources and should not attempt to make our own definition based on original research. PrimeHunter (talk) 12:34, 23 December 2023 (UTC)[reply]

It is rather clear that, for the sources and in this context, "known prime number" means "number that is known to be prime" and not "prime number that is known". Many numbers are known to be probably prime, but are not "known prime numbers" because no primality proof has ever been provided. D.Lazard (talk) 14:13, 23 December 2023 (UTC)[reply]

I'm not trying to impose any definition of "known" of my own, I'm just pointing out that the article lacks any definition of this word. If you think that is what "known" should mean, please include it in the article. We can't just include a word with a fuzzy definition in the title and just expect the reader to guess what it means. Especially when this is an article about mathematics.

There are methods of writing "the n-th prime number" with mathematical notation where n could be any number and these should obviously be excluded from the article, but at least the article should make it clear why and how these kind of constructions were excluded. Mexicochina (talk) 21:26, 23 December 2023 (UTC)[reply]

There are probably hundreds of reliable sources which have reported the largest known primes at various times. Can you point to a single one which has the type of definition you want? Or even a single source which has such a definition of a "known" prime whether it's large or not? If not then this discussion should really be over. Wikipedia is based on reliable sources and requires verifiability. We don't make up our own definitions just because one editor thinks that every single source has overlooked something important that absolutely must be included even if we have to make it up. PrimeHunter (talk) 23:51, 23 December 2023 (UTC)[reply]

Yes! That is exactly my point. No single reliable source has a definition for the word "known". That's why I made a edit explaining that "known" does not have a strict definition. But you reverted it.

Some readers might be left wondering what exactly "known" means and a quick little clarification will not hurt the article. Mexicochina (talk) 00:34, 24 December 2023 (UTC)[reply]

You have made several edits and some of them added your own unsourced definition of known prime. I guess you now refer to [3] which said:

The word "known" applied to prime numbers does not have a formal definition. This article will be listing prime numbers considered to be "known" by reliable sources.

That gives a very misleading impression that there is uncertainty about the primality of the numbers. When sources say "known" prime, nearly all of them simply mean that the number has been proved to be prime. You also want it to mean that the value of the number is "known" in some sense, e.g. by computing the digits. Reliable sources about known primes don't consider this issue at all and they have no practical reason to do it, because every sensible formal definition would give the same set of "known" values with the currently known and used primality proving methods. They all involve to make calculations on the binary expansion of the number (binary for computers, and large primes are always found by computer). I don't think anyone would seriously consider the possibility that something like "The smallest billion-digit number which is only divisible by 1 and itself" is a "known" prime just because we can prove the sentence describes a prime and we could make a formula which is equivalent of searching for it. If your edit clarified that "known" refers to the value of the number, we would still need a reliable source which says there is no formal definition. It's not good enough that we couldn't find a formal definition. PrimeHunter (talk) 02:02, 24 December 2023 (UTC)[reply]

Awesome! That explanation is a wonderful insight on the meaning of the word "known". I just wish it was included in the article for everyone to read and not just hidden in the talk page.

You agree that "known" is not a rigorously defined word, right? Then it's reasonable to expect that some people will be wondering what exactly that word means since mathematicians tend to give rigorous definitions to every word they use. If we include your explanation in the article it will become much clearer.

Almost every Wikipedia article explains the words that appear in the title. I don't see why this article has to be different. Mexicochina (talk) 05:11, 24 December 2023 (UTC)[reply]

You've made your position clear, but you haven't convinced us. "Known" is a common English word and is being used with its common meaning. Unnecessary text makes an article worse. Dan Bloch (talk) 05:43, 24 December 2023 (UTC)[reply]

I used my own knowledge of the field to give the explanation. That's not allowed in articles without a reliable source. And as mentioned, sources just mean proven primality when they say "known" prime. I think there are very few people with your concern about the meaning. If it was a common concern then sources would talk about it. If new primality tests were discovered for huge numbers where the binary expansion is infeasible to compute then I cannot say which primes would be considered "known", but it's not an issue today. PrimeHunter (talk) 12:07, 24 December 2023 (UTC)[reply]

Found Prime [Unverified]

Proof for 2⁸²⁵⁸⁹⁹³³-1 + 2,338

09-Jan-2024 Just found a new Prime cred. Wolfram Alpha and Bing Chat. 2⁸²⁵⁸⁹⁹³³-1 + 2,338 [Unverified], which has 24,862,048 digits. Professor. Damian A. James Williamson

[4]https://twitter.com/CA_22562_AN/status/1744686253705212204 KING (talk) 11:50, 9 January 2024 (UTC)[reply]

now M52 is real: https://www.mersenneforum.org/node/1055976 Just a random Wikipedian^(talk) 08:17, 13 October 2024 (UTC)[reply]

Citing references 1 and 8 (someone's private webpage) is NOT AUTH SOURCE

Landon Noll's private website is NOT an authoritative source and should be reverted. Suggest to hold on and LOCK this page until the official public release. Serge Batalov (talk) 16:30, 16 October 2024 (UTC)[reply]

If anybody is curious the two sources are no longer on the web. (And the Wikipedia changes have been rolled back.) Dan Bloch (talk) 00:55, 17 October 2024 (UTC)[reply]

graph feels like it is overfitted

suddenly changing from "doubles every x years" to "grows by 10^6 every y years" doesn't seem like it really accurately models the data Liambdonegan01 (talk) 10:10, 22 October 2024 (UTC)[reply]

Wrong Number Of Digits

The text file of the 52nd Mersenne prime has 820468 lines with 100 numbers each. And the last row has 40 characters, so it should be 82,048,640 digits long. I haven't checked all of them, but from the 52nd to the 49th Mersenne primes the number of digits seems to be half of what it is in mersenne.org, is there a reason? Turrettr (talk) 13:13, 25 October 2024 (UTC)[reply]

It seems that the lower number of digits is base 10, but I don't understand how the text file is presented or why it's so much bigger. (Hohum ^@) 15:55, 25 October 2024 (UTC)[reply]

ok, figured it out.

The new mersenne prime is about 41 million digits in base 10

https://www.mersenne.org/primes/digits/M136279841.zip

The perfect number of that prime is 82 million digits

https://www.mersenne.org/primes/perfect/perfect136279841.zip
https://www.mersenne.org/primes/?press=M136279841
"Euclid proved that every Mersenne prime generates a perfect number. A perfect number is one whose proper divisors add up to the number itself. The smallest perfect number is 6 = 1 + 2 + 3 and the second perfect number is 28 = 1 + 2 + 4 + 7 + 14. Euler (1707-1783) proved that all even perfect numbers come from Mersenne primes. The newly discovered perfect number is 2136,279,840 x (2136,279,841-1). This number is over 82 million digits long!" (Hohum ^@) 16:06, 25 October 2024 (UTC)[reply]